In this work we introduce a new two-scale reduced basis method for the LOD (TSRBLOD) that takes both the local corrector
problems as well as the coarse-scale problem into account to produce a single small-size ROM that no longer requires explicit solutions of local subproblems in the online phase.
The model is constructed based on a new two-scale formulation of the LOD as a single variational problem, inspired by
the theory of two-scale convergence in mathematical homogenization [1]}.
As the computation of solution snapshots for this model would still be computationally expensive, we combine this
formulation with a preceding reduction of the fine-scale corrector problems similar to the RBLOD approach in [2]}.
Rigorous and efficient a posteriori bounds derived from the two-scale formulation control the error over this entire
two-stage reduction process.
